3.1.67 \(\int \frac {x^2}{\text {arccosh}(a x)^4} \, dx\) [67]

3.1.67.1 Optimal result

Integrand size = 10, antiderivative size = 153 \[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {x}{3 a^2 \text {arccosh}(a x)^2}-\frac {x^3}{2 \text {arccosh}(a x)^2}+\frac {\sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \text {arccosh}(a x)}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{24 a^3}+\frac {9 \text {Chi}(3 \text {arccosh}(a x))}{8 a^3} \]

output

1/3*x/a^2/arccosh(a*x)^2-1/2*x^3/arccosh(a*x)^2+1/24*Chi(arccosh(a*x))/a^3 
+9/8*Chi(3*arccosh(a*x))/a^3-1/3*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh 
(a*x)^3+1/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/arccosh(a*x)-3/2*x^2*(a*x-1)^( 
1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)
 

3.1.67.2 Mathematica [A] (warning: unable to verify)

Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\frac {\sqrt {-1+a x} \left (-4 \sqrt {\frac {-1+a x}{1+a x}} \left (2 a^2 x^2 \left (-1+a^2 x^2\right )+a x \sqrt {-1+a x} \sqrt {1+a x} \left (-2+3 a^2 x^2\right ) \text {arccosh}(a x)+\left (2-11 a^2 x^2+9 a^4 x^4\right ) \text {arccosh}(a x)^2\right )+(-1+a x) \text {arccosh}(a x)^3 \text {Chi}(\text {arccosh}(a x))+27 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(3 \text {arccosh}(a x))\right )}{24 a^3 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^3} \]

input

Integrate[x^2/ArcCosh[a*x]^4,x]
 

output

(Sqrt[-1 + a*x]*(-4*Sqrt[(-1 + a*x)/(1 + a*x)]*(2*a^2*x^2*(-1 + a^2*x^2) + 
 a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-2 + 3*a^2*x^2)*ArcCosh[a*x] + (2 - 11* 
a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x]^2) + (-1 + a*x)*ArcCosh[a*x]^3*CoshInteg 
ral[ArcCosh[a*x]] + 27*(-1 + a*x)*ArcCosh[a*x]^3*CoshIntegral[3*ArcCosh[a* 
x]]))/(24*a^3*((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^3)
 

3.1.67.3 Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6301, 6366, 6295, 6300, 2009, 6368, 3042, 3782}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx\)

\(\Big \downarrow \) 6301

\(\displaystyle a \int \frac {x^3}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx-\frac {2 \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}dx}{3 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 6366

\(\displaystyle a \left (\frac {3 \int \frac {x^2}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {2 \left (\frac {\int \frac {1}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 6295

\(\displaystyle a \left (\frac {3 \int \frac {x^2}{\text {arccosh}(a x)^2}dx}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {2 \left (\frac {a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 6300

\(\displaystyle a \left (\frac {3 \left (-\frac {\int \left (-\frac {a x}{4 \text {arccosh}(a x)}-\frac {3 \cosh (3 \text {arccosh}(a x))}{4 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {2 \left (\frac {a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {a \int \frac {x}{\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}dx-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}+a \left (\frac {3 \left (-\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 6368

\(\displaystyle -\frac {2 \left (\frac {\frac {\int \frac {a x}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}+a \left (\frac {3 \left (-\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {x}{2 a \text {arccosh}(a x)^2}+\frac {-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}+\frac {\int \frac {\sin \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )}{\text {arccosh}(a x)}d\text {arccosh}(a x)}{a}}{2 a}\right )}{3 a}+a \left (\frac {3 \left (-\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

\(\Big \downarrow \) 3782

\(\displaystyle a \left (\frac {3 \left (-\frac {-\frac {1}{4} \text {Chi}(\text {arccosh}(a x))-\frac {3}{4} \text {Chi}(3 \text {arccosh}(a x))}{a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}\right )}{2 a}-\frac {x^3}{2 a \text {arccosh}(a x)^2}\right )-\frac {2 \left (\frac {\frac {\text {Chi}(\text {arccosh}(a x))}{a}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)}}{2 a}-\frac {x}{2 a \text {arccosh}(a x)^2}\right )}{3 a}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3}\)

input

Int[x^2/ArcCosh[a*x]^4,x]
 

output

-1/3*(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^3) - (2*(-1/2*x/(a 
*ArcCosh[a*x]^2) + (-((Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + C 
oshIntegral[ArcCosh[a*x]]/a)/(2*a)))/(3*a) + a*(-1/2*x^3/(a*ArcCosh[a*x]^2 
) + (3*(-((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) - (-1/4*Cos 
hIntegral[ArcCosh[a*x]] - (3*CoshIntegral[3*ArcCosh[a*x]])/4)/a^3))/(2*a))
 

3.1.67.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 6295

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c* 
x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c 
/(b*(n + 1))   Int[x*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + 
 c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
 

rule 6300

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + Simp[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 
 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, 
a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] 
&& LtQ[n, -1]
 

rule 6301

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1) 
)), x] + (-Simp[c*((m + 1)/(b*(n + 1)))   Int[x^(m + 1)*((a + b*ArcCosh[c*x 
])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Simp[m/(b*c*(n + 1)) 
 Int[x^(m - 1)*((a + b*ArcCosh[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]) 
), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
 

rule 6366

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1 
_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a 
 + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x 
]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Simp[f*(m/(b*c*(n + 1)))*Simp 
[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]   Int[ 
(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e 
1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

3.1.67.4 Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{24 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{24}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(121\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{24 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{24}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{12 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {\cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{8}}{a^{3}}\)	\(121\)

input

int(x^2/arccosh(a*x)^4,x,method=_RETURNVERBOSE)
 

output

1/a^3*(-1/12/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/24*a*x/arccosh(a 
*x)^2-1/24/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/24*Chi(arccosh(a*x)) 
-1/12/arccosh(a*x)^3*sinh(3*arccosh(a*x))-1/8/arccosh(a*x)^2*cosh(3*arccos 
h(a*x))-3/8/arccosh(a*x)*sinh(3*arccosh(a*x))+9/8*Chi(3*arccosh(a*x)))
 

3.1.67.5 Fricas [F]

\[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

input

integrate(x^2/arccosh(a*x)^4,x, algorithm="fricas")
 

output

integral(x^2/arccosh(a*x)^4, x)
 

3.1.67.6 Sympy [F]

\[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^{2}}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]

input

integrate(x**2/acosh(a*x)**4,x)
 

output

Integral(x**2/acosh(a*x)**4, x)
 

3.1.67.7 Maxima [F]

\[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

input

integrate(x^2/arccosh(a*x)^4,x, algorithm="maxima")
 

output

-1/6*(2*a^13*x^13 - 10*a^11*x^11 + 20*a^9*x^9 - 20*a^7*x^7 + 10*a^5*x^5 + 
2*(a^8*x^8 - a^6*x^6)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - 2*a^3*x^3 + 2*(5*a 
^9*x^9 - 9*a^7*x^7 + 4*a^5*x^5)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^10 - 
 13*a^8*x^8 + 11*a^6*x^6 - 3*a^4*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4* 
(5*a^11*x^11 - 17*a^9*x^9 + 21*a^7*x^7 - 11*a^5*x^5 + 2*a^3*x^3)*(a*x + 1) 
*(a*x - 1) + (9*a^13*x^13 - 45*a^11*x^11 + 90*a^9*x^9 - 90*a^7*x^7 + 45*a^ 
5*x^5 + (9*a^8*x^8 - 13*a^6*x^6 + 3*a^4*x^4 + a^2*x^2)*(a*x + 1)^(5/2)*(a* 
x - 1)^(5/2) - 9*a^3*x^3 + (45*a^9*x^9 - 97*a^7*x^7 + 64*a^5*x^5 - 10*a^3* 
x^3 - 2*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (90*a^10*x^10 - 258*a^8*x^8 + 264*a 
^6*x^6 - 113*a^4*x^4 + 19*a^2*x^2 - 2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2 
*(45*a^11*x^11 - 161*a^9*x^9 + 219*a^7*x^7 - 141*a^5*x^5 + 44*a^3*x^3 - 6* 
a*x)*(a*x + 1)*(a*x - 1) + (45*a^12*x^12 - 193*a^10*x^10 + 325*a^8*x^8 - 2 
70*a^6*x^6 + 112*a^4*x^4 - 19*a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a* 
x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^12*x^12 - 21*a^10*x^10 + 34*a^ 
8*x^8 - 26*a^6*x^6 + 9*a^4*x^4 - a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1) + (3 
*a^13*x^13 - 15*a^11*x^11 + 30*a^9*x^9 - 30*a^7*x^7 + 15*a^5*x^5 + (3*a^8* 
x^8 - 4*a^6*x^6 + a^4*x^4)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - 3*a^3*x^3 + ( 
15*a^9*x^9 - 31*a^7*x^7 + 20*a^5*x^5 - 4*a^3*x^3)*(a*x + 1)^2*(a*x - 1)^2 
+ (30*a^10*x^10 - 84*a^8*x^8 + 84*a^6*x^6 - 35*a^4*x^4 + 5*a^2*x^2)*(a*x + 
 1)^(3/2)*(a*x - 1)^(3/2) + 2*(15*a^11*x^11 - 53*a^9*x^9 + 71*a^7*x^7 -...
 

3.1.67.8 Giac [F]

\[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

input

integrate(x^2/arccosh(a*x)^4,x, algorithm="giac")
 

output

integrate(x^2/arccosh(a*x)^4, x)
 

3.1.67.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]

input

int(x^2/acosh(a*x)^4,x)
 

output

int(x^2/acosh(a*x)^4, x)